Theory of Sound Propagation through Ducts Carrying High-Speed Flows

Abstract

The important problem of sound propagation in ducts carrying compressible subsonic flows is analyzed on the basis that the motion is one-dimensional. The multiple reflection method is extended to the case of sinusoidally varying pressure waves and a general integral formulation is developed. It is complicated because the total reflected wave, particularly, and the total transmitted wave must depend upon the time delay incurred by the propagation of infinitesimal reflections from along the length of the duct, and this depends upon the shape of the duct. It is shown how certain exact solutions can be obtained and these are given for the wave strengths composed of wavelets having undergone single, double and triple reflections. The frequency plays the strongest role in the reflected wave, and when the singly reflected wavelets dominate, it introduces a factor [sin(Ωχ)/Ωχ]e−iΩχ on the zero-frequency reflection, Ω being proportional to the frequency, and it and χ being dependent upon the change of Mach number at the ends of the duct. In contrast the transmitted wave is hardly affected by frequency. The analytical results apply to “almost conical” ducts, either convergent or divergent with the incident wave propagating with or against the flow direction. An approximate method, based upon the analytical results is demonstrated for ducts of other form.